Edexcel A Level Further Maths: Core Pure:复习笔记5.1.1 Volumes of Revolution

What is a volume of revolution around the x-axis? 

• A solid of revolution is formed when an area bounded by a function (and other boundary equations) is rotated 360° around the x-axis
• A volume of revolution is the volume of this solid formed

Example of a solid of revolution that is formed by rotating the area bounded by the function  , the lines   and and the -axis  about the -axis

How do I find the volume of revolution around the x-axis? 

• To find the volume of revolution created when the area bounded by the function , the lines  and , and the x-axis is rotated 360° about the x-axis use the formula

• The formula may look complicated or confusing at first due to the y and dx
  • remember that y is a function of x
  • once the expression for y is substituted in, everything will be in terms of x
• π is a constant so you may see this written either inside or outside the integral
• This is not given in the formulae booklet
  • The formulae booklet does list the volume formulae for some common 3D solids – it may be possible to use these depending on what information about the solid is available

Where does the formula for the volume of revolution come from?

• When you integrate to find the area under a curve you can see the formula by splitting the area into rectangles with small widths
  • The same method works for volumes
• Split the volume into cylinders with small widths
  • The radius will be the y value
  • The width will be a small interval along the x-axis δx
• The volume can be approximated by the sum of the volumes of these cylinders

• The limit as δx goes to zero can be found by integration - just like with areas

How do I solve problems involving volumes of revolution around the x-axis? 

• Visualising the solid created is helpful
  • Try sketching some functions and their solids of revolution to help
•  STEP 1 Square y
  • Do this first without worrying about π or the integration and limits
• STEP 2 Identify the limits a and b (which could come from a graph)
• STEP 3 Use the formula by evaluating the integral and multiplying by π
  • The answer may be required in exact form (leave in terms of π)
    • If not, round to three significant figures (unless told otherwise)
• Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

What is a volume of revolution around the y-axis? 

• A solid of revolution is formed when an area bounded by a function (and other boundary equations) is rotated 360° around the y-axis
• A volume of revolution is the volume of this solid formed

Example of a solid of revolution that is formed by rotating the area bounded by the function , the lines  and  and the -axis  about the -axis

How do I find the volume of revolution around the y-axis? 

• To find the volume of revolution created when the area bounded by the function , the lines  and , and the y-axis is rotated 360° about the y-axis use the formula

• Note that although the function may be given in the form  it will first need rewriting in the form
• This is not given in the formulae booklet

How do I solve problems involving volumes of revolution around the y-axis? 

• Visualising the solid created is helpful
  • Try sketching some functions and their solids of revolution to help

• STEP 1 Rearrange into the form  (if necessary)
  • This is finding the inverse function
• STEP 2 Square x
  • Do this first without worrying about π or the integration and limits
• STEP 3 Identify the limits c and d (which could come from a graph)
• STEP 4 Use the formula by evaluating the integral and multiplying by π
  • The answer may be required in exact form (leave in terms of π)
    • If not, round to three significant figures (unless told otherwise)

• Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

What is parametric volumes of revolution?

• Solids of revolution are formed by rotating functions about the x-axis or the y-axis
• Here though, rather than given y in terms of x, both x and y are given in terms of a parameter, t
  • 
  • 
  • Depending on the nature of the functions f and g it may not be convenient or possible to find y in terms of x

How do I find volumes of revolution when x and y are given parametrically?

• The aim is to replace everything in the ‘original’ integral so that it is in terms of t
• For the ‘original’ integral   and parametric equations given in the form  use the following process
• STEP 1: Find dx or dy in terms of t and dt
  • 
• STEP 2: If necessary, change the limits from x values or y values to t values using
  •  
  • 
• STEP 3: Square y or x
  •  
  • Do this separately to avoid confusing when putting the integral together
• STEP 4: Set up the integral, so everything is now in terms of t, simplify where possible and evaluate the integral to find the volume of revolution

(if around x-axis) or  (if around y-axis)